308-203A
Introduction to Computing II
Lecture 7: Sorting 1
Fall Session 2000
Problem Definition
Sorting: Given an ordered list of elements
(x1, x2, x3, x4, … xn) on which there is an
ordering relation, rearrange them in
ascending order.
Example:
(8, 2, 9, 16, 3)
(2, 3, 8, 9, 16)
Note
We will just talk about sorting lists of
integers for demonstrative purposes, but
a more real-world application would be
to sort a database of students by alphabetical
order.
Insertion Sort
(a.k.a. Bubblesort)
Divide the list into sorted and not-yet-sorted parts.
SORTED
UNSORTED
1, 3, 7, 10, 15, 8, 78, 34, 82, 51, 17, 91, 123, 64
Each iteration: grow the sorted part by inserting
the first element from the unsorted part
Insertion Sort - pseudocode
INSERTION-SORT(A[]) ::=
{
For i := 2 to n
key := A[i]
j
:= i -1;
while (j > 0) and (A[j] > key) do
A[j+1] := A[j]
j := j - 1
A[j+1] := key
}
Order of Growth?
INSERTION-SORT(A[]) ::=
{
For i := 2 to n
key := A[i]
LOOP 1
Q(1)
j
:= i -1;
while (j >= 0) and (A[j] > key) do
A[j+1] := A[j]
Q(1)
j := j - 1
LOOP 2
Q(1)
A[j+1] := key
}
Order of Growth?
Order of Growth:
Total Time = Q ( n ) + O ( n2 + n/2)
= O ( n2 )
Proof on separate handout
Any questions?
Can We Do Better?
Definition: Divide-and-Conquer means breaking
a problem into smaller parts which can be solved
separately and then combining the solutions to
solve the original problem
We can apply a Divide-and-Conquer strategy
to the sorting problem….
Merge sort
1) Break the list in two
2) Assume we can sort the smaller lists
(recursion)
3) Merge the results together
Merge Sort Pseudocode
MERGE-SORT(A[ ])
{
if (n > 1)
MERGE-SORT(A[1 … n/2 ])
MERGE-SORT(A[ n/2 + 1 … n])
MERGE(A[1 … n])
}
Example
8, 78, 34, 82, 51, 17, 2, 91, 123, 64
1. Split
8, 78, 34, 82, 51
17, 2, 91, 123, 64
2. Recursion
8, 34, 51, 78, 82
3. Merge
2, 17, 64, 91, 123
2, 8, 17, 34, 51, 64, 78, 82, 91, 123
Order of Growth:
A Recurrence Equation
Note: Merging can be done in Q ( n )
[ can you see how??? ]
T(n) = Q ( n ) + 2 T(
n
2
)
So what does this really mean…
Solving the Recurrence
You can prove by induction that this means:
T(n) = O( n log n )
Since O( n log n ) = o ( n2 ), we’ve made headway
over Insertion-Sort
Can We Still Do Better?
No!!
O( n log n ) is optimal for the sorting problem.
This can be proven by showing that W ( n log n )
comparisons may be needed to establish the
correct order.
Any questions?